SU(2)上的一些逼近结果

SU(2)是行列式为1的2×2酉矩阵群,本文首先给出了 SU(2)上连续可微函数的 Fourier 系数的阶的估计,并通过具体例子说明所得估计式的精确程度;另外,根据α的大小,分0<α<1两种情况讨论了 Lip(α,SU(2))中函数的 Fourier 级数的收敛性情况,并对 Lip(α,SU(2))(α>0)中的类函数的 Fourier 级数的收敛性作了讨论.     

均值有界变差条件及其在Fourier分析中的应用

本文宣布了关于Fourier (三角)级数的系数数列单调性条件的一个最终推广.我们已经证明了该条件对正弦级数保持一致收敛性方面已不可作任何推广,一些Fourier分析中重要和有趣的经典定理也已经在此最终条件下重新建立.本文因而对长达90年的研究历史作了一个简要综述.这一系列论文中的第一篇原始论文公布在arXiv:math.CA/0611805 v1, November 27, 2006.     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

窗口Fourier变换反演公式的级数表示

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L2空间,本文给出重构级数收敛的充分必要条件.     

周期函数Fourier级数展开式唯一性的简单证明与推广

对于周期函数f(x)按不同的周期展开对应不同的Fourier级数,这些表面上不同的式子是否一致引起了人们的注意[1],[2].本文应用Parseval等式给出一个关于这种唯一性的简单证明,并把这一种性质推广到高维情况的多重Fourier级数.     

交错级数收敛性判别法

研究交错级数收敛性判别法.通过计算级数通项的极限和单调性得到三个判据,并对其中两个结论给出形式简化的推论,最后举例说明所提判别法的应用.     

一类广义牛顿插值级数及其应用

本文研究了一类广义牛顿插值级数的特征及收敛性问题,给出了对等距有 理插值、数值积分及二元有理插值的应用。 §1.广义牛顿插值级数的特征 1978年作者之一提出如下一类广义牛顿插值级数:已知某函数f(x)在非负整数点x=0,1,…上的值,则在区间[0, ∞)上有插值级数     

一种无穷和的定义、性质及其在马尔科夫链中的应用

对形式级数 ∑ui无穷和的定义进行了推广 ,给出蔡查罗无穷和的概念 ,讨论了它们之间的关系 .对正项级数、交错级数的蔡氏收敛性做了完整的讨论 .并将其应用于马尔科夫链的渐近性态的研究 ,得到了更为简洁、完整的结果     

L2(C,μ)中的一组Haar基及相关收敛性

本文研究经典三分Cantor集C上的平方可积空间L2(C,μ).利用投影算子的相关结论,证明了此空间上存在一组Haar型规范正交基,进而分析了L2(C,μ)中任意元素关于此基的Fourier/Haar展开,并讨论了任意元素关于此级数展开的相关收敛性.     

一类变系数迭代数列的收敛性

对由xn+1=xn+an/xpn定义的数列{xn},根据级数∑an的收敛性给出{xn}的收敛性和相应等价量,并给出多个应用例子.     

Extrapolation Algorithms for Filtering Series of Functions,and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the -algorithm) or by approximating the series by a rational function (in particular, a Padé approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.     

Convergence rate of hybrid sampling series associated with wavelets

While the convergence of the classical Fourier series has been well known, the rate of its convergence is not well acknowledged. The results regarding the rate of convergence of the Fourier series and wavelet expansions can be found in the book of Walter[5]. In this paper, we give the rate of convergence of hybrid sampling series associated with orthogonal wavelets.     

On convergence of ultraspherical lacunary series

Summary Kolmogoroff's classical result on the convergence of lacunary Fourier trigonometric series corresponding to a function of L² class has been extended to the convergence of the Fourier Ultraspherical series possessing lacunae similar to those supposed in Kolmogoroff's theorem for the trigonometric series.     

Necessary conditions for the convergence of Lebesgue-Fourier series

In this paper, necessary conditions have been investigated for the convergence of Fourier series at a point. An attempt has also been made to obtain a necessary and sufficient condition for the convergence of Fourier series of a function in a certain subspace ofL.     

A quantitative version of the Dirichlet-Jordan test for double Fourier series

For classes of functions with convergent Fourier series, the problem of estimating the rate of convergence has always been of interest. The classical theorem of Dirichlet and Jordan for functions of bounded variation assures the convergence of their Fourier series, but gives no estimate of the rate of convergence. Such an estimate was first provided by Bojani . Here we consider this problem in the case of functions of two variables that are of bounded variation in the sense of Hardy and Krause. The Dirichlet-Jordan test was first extended by Hardy from single to double Fourier series. Now, we provide a quantitative version of it. We prove our estimate in a greater generality, by introducing the so-called rectangular oscillation of a function of two variables over a rectangle.     

An asymptotically hierarchy-consistent, iterative sequence transformation for convergence acceleration of Fourier series

We derive the I transformation, an iterative sequence transformation that is useful for the convergence acceleration of certain Fourier series. The derivation is based on the concept of hierarchical consistency in the asymptotic regime. We show that this sequence transformation is a special case of the J transformation. Thus, many properties of the I transformation can be deduced from the known properties of the J transformation (like the kernel, determinantal representations, and theorems on convergence behavior and stability). Besides explicit formulas for the kernel, some basic convergence theorems for the I transformation are given here. Further, numerical results are presented that show that suitable variants of the I transformation are powerful nonlinear convergence accelerators for Fourier series with coefficients of monotonic behavior. This revised version was published online in August 2006 with corrections to the Cover Date.     

Besov spaces in theory of approximation

Summary In this paper we apply the theory of interpolation spaces to different parts of Approximation theory. We study the rate of convergence of summation processes of Fourier series and Fourier integrals. The main body of the paper is devoted to a study of the rate of convergence of solutions of difference schemes for parabolic initialvalue problems with constant coefficients and to related problems. Entrata in Redazione il 12 gennaio 1969.     

Some classes of functions and Fourier coefficients with respect to general orthonormal systems

The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.     

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejér on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.     

Infinite type homeomorphisms of the circle and convergence of Fourier series

We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.